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TRACTRIX
Curve considered by Claude Perrault in 1670, then studied
by Newton in 1676, Huygens in 1692 and Leibniz in 1693.
Other name: equitangential curve (because the tangent:T is constant along this curve) |
Differential equation: Cartesian parametrization: where Or also Cartesian equation: Transcendental curve. Curvilinear abscissa: Radius of curvature: Intrinsic equation 1: Area between the curve and the asymptote: |
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The tractrix can be defined as a tractory of the line, or, which amounts to the same thing, as a curve with constant tangent.
The initial problem posed by Claude Perrault was to find
the trajectory of a clock attached to a catenary the end of which describes
the edge of a table.
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Nowadays, the image would rather be that of the trajectory
of the back wheels of a vehicle the front wheels of which describe a line.
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The tractrix is also:
- the principal involute
of the catenary (i.e. involute
the cuspidal point of which is at the summit of the catenary); here, the
equation of the catenary is |
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- the locus of the points by which pass a tangent
to the logarithmic curve: Opposite, in blue and yellow, the two logarithmic curves, in red the corresponding tractrix, and in green the catenary, median of the two logarithmic curves. |
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- the locus of a point M constructed as
follows: point M0 describing
the logarithmic curve Ox and T the intersection of the tangent with Ox. We know that the "subtangent" TX is constant equal to a. The point M is the point of [M0T] such that MX = a; the line (MX) then wraps the tractrix. (Construction due to Pietro Milici, linked to the previous one). |
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- The locus of the centre of a hyperbolic spiral rolling without slipping on a line (it is therefore a roulette). | ![]() |
Furthermore, the orthogonal trajectories of the family of circles centred on Ox with radius a are translated tractrices. | ![]() ![]() |
The pedal of the
tractrix with respect to O is the elegant curve parametrized by looking like the regular bifolium. |
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The radial curve
of the tractrix is the
kappa.
Its rotation around the base generates the pseudo-sphere.
See also the syntractrices.
Remark: the curves with constant normal are none
other than the circles.
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© Robert FERRÉOL 2017